Average Error: 43.1 → 17.9
Time: 44.1s
Precision: 64
Internal Precision: 128
\[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le -13.938117260957034:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -2.0548727883799287 \cdot 10^{-244}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 17.120363017996606:\\ \;\;\;\;100 \cdot n + \left({\left(\left(i \cdot \frac{50}{3} + 50\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}^{\frac{1}{3}} \cdot \left(n \cdot i\right)\right) \cdot \sqrt[3]{i \cdot \frac{50}{3} + 50}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus i

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target42.5
Herbie17.9
\[100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;1 + \frac{i}{n} = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log \left(1 + \frac{i}{n}\right)}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}}\]

Derivation

  1. Split input into 4 regimes

  2. if i < -13.938117260957034

    1. Initial program 27.9

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around inf 62.9

      \[\leadsto \color{blue}{100 \cdot \frac{\left(e^{\left(\log \left(\frac{1}{n}\right) - \log \left(\frac{1}{i}\right)\right) \cdot n} - 1\right) \cdot n}{i}}\]

    3. Simplified17.6

      \[\leadsto \color{blue}{\frac{100}{\frac{i}{n}} \cdot \left({\left(\frac{i}{n}\right)}^{n} + -1\right)}\]

    if -13.938117260957034 < i < -2.0548727883799287e-244

    1. Initial program 51.2

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around 0 29.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    3. Simplified29.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}{\frac{i}{n}}\]
    4. Using strategy rm

    5. Applied div-inv29.6

      \[\leadsto 100 \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\color{blue}{i \cdot \frac{1}{n}}}\]

    6. Applied *-un-lft-identity29.6

      \[\leadsto 100 \cdot \frac{\color{blue}{1 \cdot \left(i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)\right)}}{i \cdot \frac{1}{n}}\]

    7. Applied times-frac14.9

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{1}{i} \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\frac{1}{n}}\right)}\]

    8. Applied associate-*r*15.0

      \[\leadsto \color{blue}{\left(100 \cdot \frac{1}{i}\right) \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}{\frac{1}{n}}}\]

    if -2.0548727883799287e-244 < i < 17.120363017996606

    1. Initial program 50.4

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around 0 34.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(\frac{1}{2} \cdot {i}^{2} + \frac{1}{6} \cdot {i}^{3}\right)}}{\frac{i}{n}}\]

    3. Simplified34.6

      \[\leadsto 100 \cdot \frac{\color{blue}{i + \left(i \cdot i\right) \cdot \left(\frac{1}{6} \cdot i + \frac{1}{2}\right)}}{\frac{i}{n}}\]

    4. Taylor expanded around -inf 16.0

      \[\leadsto \color{blue}{\frac{50}{3} \cdot \left({i}^{2} \cdot n\right) + \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)}\]

    5. Simplified16.0

      \[\leadsto \color{blue}{\left(i \cdot n\right) \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100 \cdot n}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt16.0

      \[\leadsto \left(i \cdot n\right) \cdot \color{blue}{\left(\left(\sqrt[3]{50 + \frac{50}{3} \cdot i} \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i}\right) \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i}\right)} + 100 \cdot n\]

    8. Applied associate-*r*16.0

      \[\leadsto \color{blue}{\left(\left(i \cdot n\right) \cdot \left(\sqrt[3]{50 + \frac{50}{3} \cdot i} \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i}\right)\right) \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i}} + 100 \cdot n\]
    9. Using strategy rm

    10. Applied pow1/316.0

      \[\leadsto \left(\left(i \cdot n\right) \cdot \left(\sqrt[3]{50 + \frac{50}{3} \cdot i} \cdot \color{blue}{{\left(50 + \frac{50}{3} \cdot i\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i} + 100 \cdot n\]

    11. Applied pow1/316.0

      \[\leadsto \left(\left(i \cdot n\right) \cdot \left(\color{blue}{{\left(50 + \frac{50}{3} \cdot i\right)}^{\frac{1}{3}}} \cdot {\left(50 + \frac{50}{3} \cdot i\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i} + 100 \cdot n\]

    12. Applied pow-prod-down16.0

      \[\leadsto \left(\left(i \cdot n\right) \cdot \color{blue}{{\left(\left(50 + \frac{50}{3} \cdot i\right) \cdot \left(50 + \frac{50}{3} \cdot i\right)\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{50 + \frac{50}{3} \cdot i} + 100 \cdot n\]

    if 17.120363017996606 < i

    1. Initial program 31.6

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\]

    2. Taylor expanded around 0 29.6

      \[\leadsto \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -13.938117260957034:\\ \;\;\;\;\left({\left(\frac{i}{n}\right)}^{n} + -1\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;i \le -2.0548727883799287 \cdot 10^{-244}:\\ \;\;\;\;\left(\frac{1}{i} \cdot 100\right) \cdot \frac{i + \left(i \cdot i\right) \cdot \left(\frac{1}{2} + i \cdot \frac{1}{6}\right)}{\frac{1}{n}}\\ \mathbf{elif}\;i \le 17.120363017996606:\\ \;\;\;\;100 \cdot n + \left({\left(\left(i \cdot \frac{50}{3} + 50\right) \cdot \left(i \cdot \frac{50}{3} + 50\right)\right)}^{\frac{1}{3}} \cdot \left(n \cdot i\right)\right) \cdot \sqrt[3]{i \cdot \frac{50}{3} + 50}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 44.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.517.98.324.260.5%
herbie shell --seed 2018354 
(FPCore (i n)
  :name "Compound Interest"

  :herbie-target
  (* 100 (/ (- (exp (* n (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) 1) (/ i n)))

  (* 100 (/ (- (pow (+ 1 (/ i n)) n) 1) (/ i n))))